The standard deviation of a sample is the square root of the variance
From the dataset, the smallest is 4 and the highest is 27
So, the range is:
[tex]Range = 27 - 4[/tex]
[tex]Range = 23[/tex]
Hence, the range is 23
The sample is given as: 6,9,9,6,9, 4, 8, 5, 8, 27
Start by calculating the mean
[tex]\bar x = \frac{\sum x}{n}[/tex]
So, we have:
[tex]\bar x = \frac{6+9+9+6+9+ 4+ 8+ 5+ 8+ 27}{10}[/tex]
[tex]\bar x = \frac{91}{10}[/tex]
[tex]\bar x = 9.1[/tex]
The population variance is then calculated as:
[tex]\sigma^2 = \frac{\sum(x - \bar x)^2}{n }[/tex]
This gives
[tex]\sigma^2 = \frac{(6 - 9.1)^2+(9- 9.1)^2+(9- 9.1)^2+(6- 9.1)^2+(9- 9.1)^2+(4- 9.1)^2+(8- 9.1)^2+(5- 9.1)^2+(8- 9.1)^2+(27- 9.1)^2}{10}[/tex]
[tex]\sigma^2 = 38.49[/tex]
Hence, the population variance is 5.5
In (a), we have:
[tex]\sigma^2 = 38.49[/tex]
Take the square roots of both sides
[tex]\sigma = 6.20[/tex]
Hence, the population standard deviation is 6.20
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